The matrix affine Poisson space (M_{ m,n}, pi_{ m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{ n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{ n(n-1)/2} on (M_{ n,n}, pi_{ n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{ n,n}, pi_{ n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{ n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete.
Categories: math.SG math.QA